BasisStatus

BasisStatus is a term used in computational linear algebra and related software to represent the assessment of a proposed basis for a vector space. It encapsulates whether a given set of vectors forms a valid basis for the intended ambient space and may include additional properties that affect subsequent computations.

A BasisStatus object typically contains indicators such as: isBasis (whether the vectors form a basis), rank (the column space rank), dimension (the dimension of the ambient space), isIndependent (linear independence), isSpanning (whether the vectors span the space), isOrthogonal and isOrthonormal (orthogonality properties), and numerical metrics such as conditionNumber. In numerical contexts, a DegeneracyReason or similar field may describe why a basis is deemed invalid, for example due to near-dependence or rank deficiency.

Computation of BasisStatus commonly involves forming a matrix whose columns are the basis vectors and performing standard tests. Rank tests determine independence and spanning, while determinant checks or evaluation of the Gram matrix assess orthogonality and orthonormality. The condition number provides a measure of numerical stability. In symbolic computations, exact arithmetic can yield definitive isBasis and isIndependent results, whereas floating-point arithmetic may require tolerance-based decisions.

Usage of BasisStatus helps libraries decide whether to proceed with vector-space operations or require refinement, such as replacing a dependent vector or adjusting the target dimension. It plays a central role in algorithms for basis construction, coordinate transformations, and numerical linear algebra.

See also: Basis, Linear independence, Span, Orthogonality, Condition number.